Solve w^4-w^2(2w_0^2+w_1^2+w_0^2)+2w_0^2w_{1}^2=0

$$w ^ { 4 } - w ^ { 2 } ( 2 w _ { 0 } ^ { 2 } + w _ { 1 } ^ { 2 } + w _ { 0 } ^ { 2 } ) + 2 w _ { 0 } ^ { 2 } w _ { 1 } ^ { 2 } = 0$$

Solve for w_0 (complex solution)

$$\left\{\begin{matrix}w_{0}=-i\left(2w_{1}^{2}-3w^{2}\right)^{-\frac{1}{2}}w\sqrt{w^{2}-w_{1}^{2}}\text{; }w_{0}=i\left(2w_{1}^{2}-3w^{2}\right)^{-\frac{1}{2}}w\sqrt{w^{2}-w_{1}^{2}}\text{, }&w\neq -\frac{\sqrt{6}w_{1}}{3}\text{ and }w\neq \frac{\sqrt{6}w_{1}}{3}\\w_{0}\in \mathrm{C}\text{, }&w=0\text{ and }w_{1}=0\end{matrix}\right.$$

Solve for w

$$w=-\frac{\sqrt{2\left(-\sqrt{9w_{0}^{4}+w_{1}^{4}-2\left(w_{0}w_{1}\right)^{2}}+3w_{0}^{2}+w_{1}^{2}\right)}}{2}$$

$$w=\frac{\sqrt{2\left(-\sqrt{9w_{0}^{4}+w_{1}^{4}-2\left(w_{0}w_{1}\right)^{2}}+3w_{0}^{2}+w_{1}^{2}\right)}}{2}$$

$$w=\frac{\sqrt{2\left(\sqrt{9w_{0}^{4}+w_{1}^{4}-2\left(w_{0}w_{1}\right)^{2}}+3w_{0}^{2}+w_{1}^{2}\right)}}{2}$$

$$w=-\frac{\sqrt{2\left(\sqrt{9w_{0}^{4}+w_{1}^{4}-2\left(w_{0}w_{1}\right)^{2}}+3w_{0}^{2}+w_{1}^{2}\right)}}{2}$$

Solve for w_0

$$\left\{\begin{matrix}w_{0}=\sqrt{\frac{w^{2}\left(w_{1}^{2}-w^{2}\right)}{2w_{1}^{2}-3w^{2}}}\text{; }w_{0}=-\sqrt{\frac{w^{2}\left(w_{1}^{2}-w^{2}\right)}{2w_{1}^{2}-3w^{2}}}\text{, }&\left(w\leq -|w_{1}|\text{ and }w<-\frac{\sqrt{6}|w_{1}|}{3}\right)\text{ or }\left(w\geq |w_{1}|\text{ and }w>\frac{\sqrt{6}|w_{1}|}{3}\right)\text{ or }\left(|w|\leq |w_{1}|\text{ and }|w|<\frac{\sqrt{6}|w_{1}|}{3}\right)\text{ or }\left(|w|=|w_{1}|\text{ and }|w|\neq \frac{\sqrt{6}|w_{1}|}{3}\right)\\w_{0}\in \mathrm{R}\text{, }&w=0\text{ and }w_{1}=0\end{matrix}\right.$$